Related papers: Two-level deep domain decomposition method

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Neural network-driven domain decomposition for efficient solutions to the Helmholtz equation [0.0]
Accurately simulating wave propagation is crucial in fields such as acoustics, electromagnetism, and seismic analysis.<n>Traditional numerical methods, like finite difference and finite element approaches, are widely used to solve governing partial differential equations (PDEs) such as the Helmholtz equation.<n>This work investigates Finite Basis Physics-Informed Neural Networks (FBPINNs) and their multilevel extensions as a promising alternative.
arXiv Detail & Related papers (2025-11-19T13:58:32Z)
Unrolled Graph Neural Networks for Constrained Optimization [83.29547301151177]
We study the dynamics of the dual ascent algorithm in two coupled graph neural networks (GNNs)<n>We propose a joint training scheme that alternates between updating the primal and dual networks.<n>Our numerical experiments demonstrate that our approach yields near-optimal near-feasible solutions.
arXiv Detail & Related papers (2025-09-21T16:55:41Z)
Recursive Bound-Constrained AdaGrad with Applications to Multilevel and Domain Decomposition Minimization [0.0]
Two OFFO noise tolerant algorithms are presented that handle bound constraints, inexact gradients and use second-order information.<n> numerical experiments are discussed on applications ranging from PDE-based problems to deep neural network training, illustrating their remarkable computational efficiency.
arXiv Detail & Related papers (2025-07-15T17:32:10Z)
Non-overlapping, Schwarz-type Domain Decomposition Method for Physics and Equality Constrained Artificial Neural Networks [0.24578723416255746]
We present a non-overlapping, Schwarz-type domain decomposition method with a generalized interface condition. Our approach employs physics and equality-constrained artificial neural networks (PECANN) within each subdomain. A distinct advantage our domain decomposition method is its ability to learn solutions to both Poisson's and Helmholtz equations.
arXiv Detail & Related papers (2024-09-20T16:48:55Z)
General-Kindred Physics-Informed Neural Network to the Solutions of Singularly Perturbed Differential Equations [11.121415128908566]
We propose the General-Kindred Physics-Informed Neural Network (GKPINN) for solving Singular Perturbation Differential Equations (SPDEs) This approach utilizes prior knowledge of the boundary layer from the equation and establishes a novel network to assist PINN in approxing the boundary layer. The research findings underscore the exceptional performance of our novel approach, GKPINN, which delivers a remarkable enhancement in reducing the $L$ error by two to four orders of magnitude compared to the established PINN methodology.
arXiv Detail & Related papers (2024-08-27T02:03:22Z)
Compositional Curvature Bounds for Deep Neural Networks [7.373617024876726]
A key challenge that threatens the widespread use of neural networks in safety-critical applications is their vulnerability to adversarial attacks. We study the second-order behavior of continuously differentiable deep neural networks, focusing on robustness against adversarial perturbations. We introduce a novel algorithm to analytically compute provable upper bounds on the second derivative of neural networks.
arXiv Detail & Related papers (2024-06-07T17:50:15Z)
Implicit Stochastic Gradient Descent for Training Physics-informed Neural Networks [51.92362217307946]
Physics-informed neural networks (PINNs) have effectively been demonstrated in solving forward and inverse differential equation problems. PINNs are trapped in training failures when the target functions to be approximated exhibit high-frequency or multi-scale features. In this paper, we propose to employ implicit gradient descent (ISGD) method to train PINNs for improving the stability of training process.
arXiv Detail & Related papers (2023-03-03T08:17:47Z)
Neural Basis Functions for Accelerating Solutions to High Mach Euler Equations [63.8376359764052]
We propose an approach to solving partial differential equations (PDEs) using a set of neural networks. We regress a set of neural networks onto a reduced order Proper Orthogonal Decomposition (POD) basis. These networks are then used in combination with a branch network that ingests the parameters of the prescribed PDE to compute a reduced order approximation to the PDE.
arXiv Detail & Related papers (2022-08-02T18:27:13Z)
A coarse space acceleration of deep-DDM [0.0]
We present an extension of the recently proposed deep-ddm approach to solving PDEs. We show that coarse space correction is able to alleviate the deterioration of the convergence of the solver. Experimental results demonstrate that our approach induces a remarkable acceleration of the deep-ddm method.
arXiv Detail & Related papers (2021-12-07T14:41:28Z)
Optimization-Based Separations for Neural Networks [57.875347246373956]
We show that gradient descent can efficiently learn ball indicator functions using a depth 2 neural network with two layers of sigmoidal activations. This is the first optimization-based separation result where the approximation benefits of the stronger architecture provably manifest in practice.
arXiv Detail & Related papers (2021-12-04T18:07:47Z)
Lower Bounds and Optimal Algorithms for Smooth and Strongly Convex Decentralized Optimization Over Time-Varying Networks [79.16773494166644]
We consider the task of minimizing the sum of smooth and strongly convex functions stored in a decentralized manner across the nodes of a communication network. We design two optimal algorithms that attain these lower bounds. We corroborate the theoretical efficiency of these algorithms by performing an experimental comparison with existing state-of-the-art methods.
arXiv Detail & Related papers (2021-06-08T15:54:44Z)
Solving Sparse Linear Inverse Problems in Communication Systems: A Deep Learning Approach With Adaptive Depth [51.40441097625201]
We propose an end-to-end trainable deep learning architecture for sparse signal recovery problems. The proposed method learns how many layers to execute to emit an output, and the network depth is dynamically adjusted for each task in the inference phase.
arXiv Detail & Related papers (2020-10-29T06:32:53Z)
Communication-Efficient Distributed Stochastic AUC Maximization with Deep Neural Networks [50.42141893913188]
We study a distributed variable for large-scale AUC for a neural network as with a deep neural network. Our model requires a much less number of communication rounds and still a number of communication rounds in theory. Our experiments on several datasets show the effectiveness of our theory and also confirm our theory.
arXiv Detail & Related papers (2020-05-05T18:08:23Z)
PFNN: A Penalty-Free Neural Network Method for Solving a Class of Second-Order Boundary-Value Problems on Complex Geometries [4.620110353542715]
We present PFNN, a penalty-free neural network method, to solve a class of second-order boundary-value problems. PFNN is superior to several existing approaches in terms of accuracy, flexibility and robustness.
arXiv Detail & Related papers (2020-04-14T13:36:14Z)

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